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Using the Chuang and Nielsen Quantum Computation and Quantum Information book, for an introductory course in Quantum Information, I stumbled on an ambiguous (at least for me) use of the Von Neumann entropy, being used sometimes for quantum state (for example $S(\rho)$) and sometimes for quantum systems:

Suppose distinct quantum systems A and B have a joint state ρAB. Then the joint entropy for the two systems satisfies the inequalities $$ S(A,B) \le S(A)+SA(B) $$

Sometimes it is said that $S(A)$ is the entropy of a state in the system $A$, but the example cited above seems not the case.

My question is, what is the meaning of the Von Neumann entropy of a system? Does it always is related to the entropy of a state in the aforementioned system or has it a different meaning?

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  • $\begingroup$ what do you mean by "system" as opposed to "state"? $\endgroup$ Commented Sep 28, 2018 at 9:13

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Maximize over all states $\rho$ on A. For the classical example, one bit can be in two states but we have not stated the probabilities. The maximum entropy is when $p=1/2$. So by an abuse of notation we can call it 1 bit of information even if it might really store less information.

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